Projection-difference method with the Crank-Nicolson scheme in time for the approximate solution of a parabolic equation with an integral condition for the solution

2015 ◽  
Vol 51 (1) ◽  
pp. 116-126 ◽  
Author(s):  
V. V. Smagin
Keyword(s):  
Approximate Solution ◽  
Parabolic Equation ◽  
Difference Method ◽  
Integral Condition ◽  
Nicolson Scheme ◽  
Crank Nicolson

scholarly journals A New Alternating Segment Crank-Nicolson Scheme for the Fourth-Order Parabolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Ge-yang Guo ◽  
Bo Liu
Keyword(s):  
Parabolic Equation ◽  
Fourth Order ◽  
Unconditional Stability ◽  
Truncation Errors ◽  
Order Parabolic Equation ◽  
Intrinsic Parallelism ◽  
Fourth Order Parabolic Equation ◽  
The Stability ◽  
Nicolson Scheme ◽  
Crank Nicolson

A group of asymmetric difference schemes to approach the fourth-order parabolic equation is given. According to these schemes and the Crank-Nicolson scheme, an alternating segment Crank-Nicolson scheme with intrinsic parallelism is constructed. The truncation errors and the stability are discussed. Numerical simulations show that this new scheme has unconditional stability and high accuracy and convergency, and it is in preference to the implicit scheme method.

ON THE CONVERGENCE AND RATE OF THE CONVERGENCE OF A PROJECTION-DIFFERENCE METHOD FOR APPROXIMATE SOLVING A PARABOLIC EQUATION WITH WEIGHT INTEGRAL CONDITION

2018 ◽  
pp. 517-523
Author(s):  
Anastasiya Alexandrovna Petrova
Keyword(s):  
Finite Element Method ◽  
Hilbert Space ◽  
Approximate Solution ◽  
Parabolic Equation ◽  
Integral Condition ◽  
Approximate Solutions ◽  
Linear Parabolic Equation ◽  
The Finite Element Method ◽  
Spatial Variables ◽  
Euler’S Method

In the Hilbert space the abstract linear parabolic equation with nonlocal weight integral condition for the solution is resolved approximately by projectiondifference method using time-implicit Euler’s method. Approximation of the problem by spatial variables is oriented on the finite element method. Errors estimations of approximate solutions, convergence of approximate solution to exact one and orders of rate of convergence are established.

A parallel Crank–Nicolson finite difference method for time-fractional parabolic equation

2014 ◽  
Vol 22 (4) ◽  
Author(s):  
N. H. Sweilam ◽  
H. Moharram ◽  
N. K. Abdel Moniem ◽  
S. Ahmed
Keyword(s):  
Parabolic Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Cluster Computing ◽  
Performance Metrics ◽  
Numerical Test ◽  
Preconditioned Conjugate Gradient ◽  
Pc Cluster ◽  
Crank Nicolson

Abstract- In this paper, a parallel Crank-Nicolson finite difference method (C-N-FDM) for time-fractional parabolic equation on a distributed system using MPI is investigated. The fractional derivative is described in the Caputos sense. The resultant large system of equations is studied using preconditioned conjugate gradient method (PCG), with the implementation of cluster computing on it. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication. To examine the efficiency and accuracy of the proposed method, numerical test experiment using different number of nodes of the Linux PC cluster is studied. The performance metrics clearly show the benefit of using the proposed approach on the Linux PC cluster in terms of execution time reduction and speedup with respect to the sequential running in a single PC.

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